cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A098385 Ordered factorizations over hook-type prime signatures with exactly three distinct primes (third column of A098348).

Original entry on oeis.org

13, 44, 132, 368, 976, 2496, 6208, 15104, 36096, 84992, 197632, 454656, 1036288, 2342912, 5259264, 11730944, 26017792, 57409536, 126091264, 275775488, 600834048, 1304428544, 2822766592, 6090129408, 13103005696, 28118614016, 60196651008
Offset: 0

Views

Author

Alford Arnold, Sep 06 2004

Keywords

Comments

a(n) can also be calculated by transforming (3,18,8) applying the binomial transform twice. Cf. A098384.

Examples

			The hook-type least prime signatures with exactly three primes begin 30,60,120,...; therefore sequence begins A002033(30,60,120,...) = 13,44,132,...

Links

Crossrefs

Cf. A002033, A098348, A098384.

Programs

  • Python
    def a(n): return 2**n * (n**2 + 8*n + 13) # James Rayman, Mar 27 2021

Formula

a(n) = 2^n * (n^2 + 8*n + 13). - James Rayman, Mar 27 2021

Extensions

More terms from James Rayman, Mar 27 2021

A098384 Triangle read by rows of coefficients used to generate diagonals of ordered factorizations as displayed in A098348.

Original entry on oeis.org

1, 3, 2, 13, 18, 8, 75, 158, 144, 48, 541, 1530, 2120, 1440, 384, 4683, 16622, 30960, 31920, 17280, 3840
Offset: 0

Views

Author

Alford Arnold, Sep 06 2004

Keywords

Comments

Note that the table includes the well-known sequence (A000165) discussed by Gordon on pages 636-645 of AMM 106 (1999).

Examples

			The table begins:
1
3 2
13 18 8
75 158 144 48
541 1530 2120 1440 384
The binomial transform of (13,18,8) yields 13,31,57,91,...
The binomial transform of 13,31,57,91,... yields 13,44,132,368,... A098385

Crossrefs

Cf. A000165, A052876, A098348, A098385. A194649.

Formula

From Peter Bala, Apr 20 2012: (Start)
The following formulas are all conjectural:
T(n,k) = 2^k*sum {i = k+1..n+1} binomial(i,k+1)*(i-1)!*Stirling2(n+1,i) = 1/(k+1)*A194649(n+1,k).
Recurrence equation:
T(n,k) = 2*k*T(n-1,k-1) + 3*(k+1)*T(n-1,k) + (k+2)*T(n-1,k+1).
E.g.f.: exp(x)/((2-exp(x))*(2*t+2-(2*t+1)*exp(x))) = 1 + (3+2*t)*x + (13+18*t+8*t^2)*x^2/2! + ....
Column n generating function: 2^n*exp(x)*(1-exp(x))^n/(exp(x)-2)^(n+2) for n >= 0.
(End)

A098349 Sum of ordered factorizations over hook-type prime signatures. (Row sums of A098348).

Original entry on oeis.org

1, 5, 25, 147, 1045, 8883, 88389, 1008723, 12984933, 186065011, 2936648325, 50611122451, 945623586725, 19037956164787, 410846314815941, 9460698944482643, 231538646070940901, 6001404380246870771, 164229726267373422853, 4731532138668375166355, 143154470312498479646245, 4538018949359464639487283, 150411282096757092041331781
Offset: 1

Views

Author

Alford Arnold, Sep 03 2004

Keywords

Comments

The general case for all prime signatures is described by A095705 with row sums A035341.

Examples

			The array begins
1
2 3
4 8 13
8 20 44 75
16 48 132 308 541
...
therefore sequence begins 1 5 25 147 1045 ...

Crossrefs

Cf. A035341, A095705, A098348.

Extensions

More terms from David Wasserman, Feb 20 2008

A131420 A tabular sequence of arrays counting ordered factorizations over least prime signatures. The unordered version is described by sequence A129306.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 26, 16, 48, 132, 308, 541, 76, 176, 32, 112, 368, 1076, 2612, 4683, 208, 604, 1460, 252, 818, 64, 256, 876, 3408, 10404, 25988, 47293, 544, 1888, 5740, 14300, 768, 2316, 3172, 7880, 128, 576, 2496, 10096, 36848, 116180
Offset: 1

Views

Author

Alford Arnold, Jul 10 2007

Keywords

Comments

The display has 1 2 3 5 7 11 15 ... terms per column. (cf. A000041)
The arrays begin
1.....2.....4......8......16.....32.....64......128
......3.....8.....20......48....112....256......576
...........13.....44.....132....368....976.....2496
..................75.....308...1076...3408....10096
.........................541...2612..10404....36848
...............................4683..25988...116180
.....................................47293...296564
.............................................545835
..................26......76....208....544
.........................176....604...1888
...............................1460...5740
.....................................14300
................................252....768
......................................2316
................................818...3172
......................................7880
with column sums
1....5....25....173....1297....12225....124997 => A035341
Column i corresponds to partitions of i. The rows correspond successively to the partitions {i}, {i-1,1},{i-2,1,1},{i-3,1,1,1}, ..., {i-7,1,1,1,1,1,1,1}, {i-2,2}, {i-3,2,1}, {i-4,2,1,1}, {i-5,2,1,1,1}, {i-3,3}, {i-3,3,1}, {i-4,2,2}, {i-5,2,2,1}. - Roger Lipsett, Feb 26 2016

Examples

			36 = 2*2*3*3 and is in A025487. There are 26 ways to factor 36 so a(11) = 26.

Crossrefs

Cf. A000041, A000670, A002033, A025487, A035098, A035341, A050324, A074206, A095705, A098348, A104725, A108464, A129306.

Programs

  • Mathematica
    gozinta counts ordered factorizations of an integer, and if lst is a partition we have
gozinta[1] = 1;
gozinta[n_] := gozinta[n] = 1 + Sum[gozinta[n/i], {i, Rest@Most@Divisors@n}]
a[lst_] := gozinta[Times @@ (Array[Prime, Length@lst]^lst)] (* Roger Lipsett, Feb 26 2016 *)

Extensions

Corrected entries in table in comments section - Roger Lipsett, Feb 26 2016
Showing 1-4 of 4 results.

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